Tagged Questions

A groupoid (in the sense of category theory) is a small category in which every morphism is an isomorphism. Groupoids arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical ...

Given a 2-groupoid $G$, two objects $a,b$, and two 1-morphisms $f,g:a\rightarrow b$ and a 2-morphism $\alpha : f \rightarrow g$, is it the case that
there always exists a 2-morphism $\beta : f^{-1} ...

It is stated in J.P.May's A Concise Course in Algebraic Topology page 29 that the fundamental groupoid functor induces a bijection
$$Cov(E,E')\longleftrightarrow Cov(\Pi(E),\Pi(E')).$$
So does that ...

I'm newbie to the category and groupoid, and I got confused about the definition of groupoids.
In the definition of groupoid in the Wikipedia, it says a groupoid is a "small" category such that every ...

For Hausdorff topological groups, the set $\{e\}$ containing only the identity is closed. This is because Hausdorff implies T1 which implies singletons are closed.
For topological groupoids, defined ...

If I have a particularly nice space $X$ (Hausdorf, locally path connected, semi-locally 1-connected, I think), then then there is an equivalence of categories between the cover category over $X$ and ...

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). ...

One may readily show that a connected groupoid $G$ is determined up to isomorphism by a group (one of the groups $\hom_G(x,x)$) and by a set (the set of all objects). This is the nature of the problem ...

I'm trying to understand the Homotopy Type Theory book. I find myself completely lost in Chapter 2, especially when it starts using higher groupoids.
What is the recommended background for this book? ...

Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$
there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$
if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...

Are there two path connected topological spaces $X,Y$ such that the fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$ but the fundamental group of $X$ is isomorphic to ...

It is written in Wikipedia http://en.wikipedia.org/wiki/Groupoid,
that any connected groupoid $A\rightrightarrows X$ is isomorphic to the action groupoid
coming from a transitive action of some group ...

This is a question about the definition of a groupoid (in category theory).
I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...

I've recently encounted a problem in my reading which would seem to be more naturally phrased if the category we work in shifted from the category $\textbf{Top}^*$ of pointed topological spaces, to ...

Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category.
First can someone tell me ...

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton .
Suppose we have a span of span of groupoids as follows and ...

For some reason I convinced myself that a simplicial set (or maybe I mean directly Kan complex) is homotopy equivalent to the nerve of a groupoid if and only if it has no higher homotopy groups.
Is ...

You will have to forgive me as I am very new to category theory - fifth of the way through Categories for a working mathematician. I'm interested in the following;
Let $F:A \to B$ and $G:A \to C$ be ...